Operator Entanglement from Non-Commutative Symmetries

Abstract: We argue that Hopf-algebra deformations of symmetries -- as encountered in non-commutative models of quantum spacetime -- carry an intrinsic content of $operator$ $entanglement$ that is enforced by the coproduct-defined notion of composite generators. As a minimal and exactly solvable example, we analyze the $U_q(\mathfrak{su}(2))$ quantum group and a two-qubit realization obtained from the coproduct of a $q$-deformed single-spin Hamiltonian. Although the deformation is invisible on a single qubit, it resurfaces in the two-qubit sector through the non-cocommutative coproduct, yielding a family of intrinsically nonlocal unitaries. We compute their operator entanglement in closed form and show that, for Haar-uniform product inputs, their entangling power is fully determined by the latter. This provides a concrete mechanism by which non-commutative symmetries enforce a baseline of entanglement at the algebraic level, with implications for information dynamics in quantum-spacetime settings and quantum information processing.